SIMILAR SURFACES.

460. Figures which have the same form and differ only in size are Similar Figures.

The following truths regarding similar figures can be established by geometry:

461. PRINCIPLES.

1. Similar surfaces are to each other

as the squares of their corresponding dimensions.

2. The corresponding dimensions of similar surfaces are to each other as the square roots of their areas.

1. There are two circular gardens, one having a diameter of 8 rods and the other 32 rods. How do they compare in size?

SOLUTION. Since the gardens are similar in form, their sizes or areas are to each other as the squares of the same dimensions in each; that is, as 82 is to 322, or as 64 is to 1024. Since 1024 is 16 times 64, the larger garden is, therefore, 16 times as large as the smaller.

2. A lady had a circular flower-bed 8 feet in diameter, and a similar one four times as large. What was the diameter of the larger bed?

3. A has a rectangular field 80 rods long and 60 rods wide. What will be the dimensions of a similar field containing 131 acres?

4. If a horse tied to a stake by a rope 7.13 rods in length can graze upon just one acre of ground, how long should the rope be to allow him to graze upon 61 acres?

5. If 6 gallons of water flow through a pipe 1 inch in diameter in a minute, how many gallons will flow through a pipe 4 inches in diameter in 5 minutes, when the stream moves with the same velocity?

6. A school-room contained two square blackboards whose sides were 3 ft. and 6 ft. respectively. How did they compare in area?

462. 1. How many figures are required to express the cube of any number of units?

2. How does the number of figures required to express the cube of any number between 9 and 100 compare with the number of figures expressing the number?

3. How does the number of figures expressing the cube of any number between 99 and 1000 compare with the number of figures expressing the number?

4. If, then, the cube of a number is expressed by 4 figures, how many orders of units are there in the root? If by 5 figures, how many? If by 6 figures, how many? figures, how many?

If by 8

5. How may the number of figures in the cube root of a number be found?

463. PRINCIPLES.-1. The cube of a number is expressed by three times as many figures as the number itself, or by one or two less than three times as many.

2. The orders of units in the cube root of a number correspond to the number of periods of three figures each into which the number can be separated, beginning at units.

The left-hand period may contain one, two, or three figures.

464. If the tens of a number are represented by t and the units by u, the cube of a number consisting of tens and units will be the cube of (t+u) or t+3tu + 3 tu+u3, Art. 449.

Thus, 353 tens + 5 units, or 30+ 5, and 353 = 303 + 3(302 × 5) + 3(30 × 52) + 58 = 42875.

WRITTEN EXERCISES.

465. 1. What is the cube root of 13824, or what is the edge of a cube whose solid contents are 13824 units?

13-824(20+4=24

EXPLANATION.-According to Prin. 2, Art. 463, the orders of units in the cube root of any number may be determined from the number of periods obtained by separating the number into periods containing three Separating the given number thus, is composed of tens and units.

The tens in the cube root of the number cannot be greater than 2, for the cube of 3 tens is 27000. 2 tens, or 20 cubed, are 8000, which, subtracted from 13824, leave 5824; therefore the root, 20, must be increased by a number such that the additions will exhaust the remainder.

The cube (A) already formed from the 13824 cubic units is one whose edge is 20 units. The additions to be made, keeping the figure formed a perfect cube, are 3 equal rectangular solids, B, C, and D; 3 other equal rectangular solids, E, F, and G; and a small cube, H. Inasmuch as the solids, B, C, and D, comprise much the greatest part of the additions, their solid contents will be nearly 5824 cubic units, the contents of the addition.

Since the cubical contents of these three equal solids are nearly equal to 5824 units, and the superficial contents of a side of each of these solids are 20 x 20, or 400 square units, if we divide 5824 by 3 times 400, or 1200, since there are 3 equal solids, we shall obtain the thickness of the addition, which is 4 units.

Since all the additions have the same thickness, if their superficial contents, or area of each side, are multiplied by 4, the result will be the solid contents of these

additions.

Besides the larger additions there are three others, E, F, and G, which are each 20 units long and 4 units wide, or whose surfaces have an area of 80 units each, or 240 units altogether; and a small cube whose sides have an area of 16 units. The sum of these areas, 1456, multiplied by 4, the thickness of the additions, gives the solid contents of the additions, which are 5824 units.

Therefore the edge of the cube is 24 units in length, or the cube root of 13824 is 24.

EXPLANATION. In the same manner as before, it may be shown that the root of the num

ber contains only tens and units.

The tens cannot be greater than 2, for 3 tens raised to the third power is 27000. Cubing 2 tens and subtracting, there is left 5824. This remainder contains 3 times the tens2 x the units + 3 times the tens x the units2, + the 13.824 (24 units3.

ts=203= 8 000 3t2=202x3=1200 5 824

3txu= (20x4) x3= 240

u2=4x4= 16

312+3 tu+u2=1456

Each of these parts contains the units as a factor, hence 5284 is the product of two factors, one of which is the units, and the other 3 times the tens2 + 3 times the tens x the units + the units2.

Since 3 times the tens2 is much greater than the rest of the factor, if 5284 is divided by 3 times the tens2, or 1200, the quotient will be the units or other factor. It is found to be 4.

The factor completed is therefore 3 x 202 + 3 x 20 x 4 + 42, which is equal to 1200 +240 +16, or 1456. This multiplied by 4 gives the product 5824. Therefore the cube root of the number is 24.

STAND. AR.-22

When the number consists of more than two orders of units, the root may be found in the same manner by considering each time the root already found as tens and the next order of the root as units.

When the number of figures in the root is more than two the following method materially abridges the process: